Continuity and Discontinuity

Hien Pham and Kim Lai

Calculus relies heavily upon the Calculus relies heavily upon the existence of what’s called continuous existence of what’s called continuous

functions. In fact, most of the functions. In fact, most of the important theorems require that any important theorems require that any

function in question must be function in question must be continuous before one can even think continuous before one can even think

about applying the theorems.about applying the theorems.

ImportanceImportance

ContinuityContinuity

““Given a continuous function Given a continuous function f(x)f(x)…”…”

““As long as As long as g(x) g(x) is a continuous function…is a continuous function…””

““Assume Assume h(x) h(x) is continuous on [a,b]…”is continuous on [a,b]…”

Continuous functions are predictable…

No breaks in the graph No holes

No jumps

A function such as g(x)= has a discontinuity at x=3 because the denominator is zero there

3

92

x

x

It is reasonable to say that the function is “continuous” everywhere else because the

graph seems to have no other “gaps” or “jumps”

Continuity

Continuity at a Point:

A function f(x) is said to be continuous x = c if each of the following conditions is satisfied:

)(lim xfcx

)(lim xfcx

1. f(c) exists,

2. exists, and

3. = f(c)

Continuity

Continuity on an Interval: A function f(x) is continuous on an interval of x-

values if and only if it is continuous at each value of x in that interval

At the end points of a closed interval, only the one-sided limits need to equal the function

Cusp A cusp is a point on the graph at which the

function is continuous but the derivative is discontinuous

Verbally: A cusp is a sharp point or an abrupt change in direction

The graph can have a cusp (an abrupt change in direction) at x=c and still be

continuous there

Removable Discontinuities

cxlim f(x) = L exists (and is finite)

but f(c) is not defined or f(c) ≠ L

You can define or redefine the value of f(c) to make f continuous at this point

cxlim f(x) = L but f(c) is not definedIf

then the discontinuity at x=c can be removed by defining f(c) = L

ExampleExample

We can “remove” the discontinuity by filling the hole

)1(

)1( 2

x

xConsider the function g(x) = . Then g(x) = (x + 1) for all real numbers except x=1

Since g(x) and x+1 agree all points other than the objective, 21lim)(lim11

xxgxx

The domain of g(x) may be extended to include x=1 by declaring that g(1)=2. This makes g(x) continuous at x=1. Since g(x) is continuous at all other points by

defining g(x)=2 turns g into a continuous function.

Lxfcx

)(limIf but f(a) ≠ L

then the discontinuity at x=a can be removed by redefining f(a)=L

ExampleExample

25.0

15.1

,3

,

)(

x

Undefined

xhConsider the functionUnless 0<x<1

If x=0.5

0<x<1, x≠0.5

We can remove the discontinuity by redefining the function so as to fill the hole

In this case, we redefine h(0.5)=1.5 + (1/.75) = 17/6

Step DiscontinuityAlthough there is a value for f(c), f(x)

approaches different values from the left of c and the right of c. So, there is no limit

of f(x) as x approaches c.

You cannot remove a step discontinuity simply by redefining f(c)

Example

Infinite DiscontinuityAs x gets closer to c, the value of f(x)

becomes large without bound.

The discontinuity is not removable just by redefining f(c)

Example

The graph approaches a vertical asymptote at x = c

One-sided Limits and Piecewise One-sided Limits and Piecewise FunctionsFunctions

The graph is an example of a The graph is an example of a function that has different one-function that has different one-sided limits as sided limits as xx approaches approaches cc..

-As -As xx approaches c from the left approaches c from the left side, side, f(x)f(x) stays close to 4. stays close to 4.

-As -As xx approaches c from the right approaches c from the right side, side, f(x)f(x) stays close to 7. stays close to 7.

One-sided Limits

)(lim xfcx

)(lim xfcx

)(lim xfLcx

x c from the left (through values of x on the negative side of c)

x c from the right (through values of x on the positive side of c)

)(lim xfLcx

)(lim xfLcx

if and only if and

A step discontinuity can result if f(x) is defined by a different rule for c than it is for the piece to the left

Each part of the function is called a branch. You can plot the three branches on your grapher by

entering the three equations, then dividing by the appropriate Boolean variable.

A Boolean variable, such as (x ≤ 2), equals 1 if the condition inside the parenthesis is true and 0 if the condition is false.

Piecewise Function

if x ≤ 2

if 2 < x < 5

if x ≥ 5

,2

,88

,4

)( 2

x

xx

x

xf

Example 1Example 1

For the piecewise function f shown,

a. Does f(x) have a limit as x approaches 2? Explain. Is f continuous at x = 2?

b. Does f(x) have a limit as x approaches 5? Explain. Is f continuous at x = 5?

if x ≤ 2

if 2 < x < 5

if x ≥ 5

,2

,88

,4

)( 2

x

xx

x

xf

Solution (part a)Solution (part a)

The function The function ff is discontinuous at x = 2. is discontinuous at x = 2.

6)(lim2

xfx

4)(lim2

xfx

)(lim2

xfx

and

does not exist.

The left and right limits are unequal.

There is a step discontinuity.

Solution (part b)Solution (part b)

The function The function ff is continuous at x = 5 because is continuous at x = 5 because the limit as the limit as xx approaches 5 is equal to the approaches 5 is equal to the

function value at 5.function value at 5.

7)(lim5

xfx

)5(7)(lim5

fxfx

7)(lim5

xfx

The open circle at the right end of the middle branch is filledwith the closed dot on the left end of the right branch.

The left and right limits are equal.and

Example 2Example 2

a. Find the value of k that makes the function continuous at x = 2.

b. Plot and sketch the graph.

,43

,)(

2

x

kxxh

if x < 2

if x ≥ 2Let the function

Solution (part a)Solution (part a)

For For hh to be continuous at x = 2, the two to be continuous at x = 2, the two limits must be equal.limits must be equal.

kkxhx

42)(lim 2

2

5432)(lim2

xhx

25.154 kk

Solution (part b)Solution (part b)

The missing point at the end of the left branch is The missing point at the end of the left branch is filled by the point at the end of the right branch, filled by the point at the end of the right branch,

showing graphically that showing graphically that hh is continuous at x = 2. is continuous at x = 2.

The End